How to Make It
Put the mushrooms in a bowl and cover with the hot water. Let stand until softened, about 10 minutes. Lift the mushrooms out of the water and rinse. Cut any large mushrooms in half lengthwise.
Cut the white of each leek crosswise into 1/3-inch-thick slices; keep the rings intact. Thinly slice enough leek greens to make 1/2 cup; reserve the rest for another use.
In a small saucepan of simmering water, cook the leek greens for 1 minute; drain well. In a small skillet, melt 1/2 tablespoon of the butter. Add the leek greens and cook over moderate heat for 1 minute, stirring. Transfer the greens to a blender. Add 2 tablespoons of water and blend until smooth; season the puree with salt.
In a large, deep skillet, bring 6 cups of water to a boil with the thyme and remaining 1 tablespoon of butter. Season with salt, cover and cook for 2 minutes. Uncover and add the sliced leeks. Simmer over moderately low heat until tender, 12 minutes. Line a large baking sheet with paper towels. With a slotted spoon, transfer the leeks to the paper towels to drain; pat dry.
In a small bowl, mix 1/4 cup of the olive oil with the vinegar and season with salt and pepper. In a small skillet, heat the remaining 1 tablespoon of olive oil. Add the mushrooms, season with salt and pepper and toss over moderately high heat until warmed through, about 1 minute.
Dollop the pureed leek greens onto a platter. Arrange the leeks on the platter and spoon the vinaigrette on top. Scatter with the mushrooms and mint and serve.